F_{n} = \frac{1}{\sqrt{5}} (α^{n} - β^{n})

where

α

and

β

are solutions of the quadratic equation

x^{2} - x - 1 = 0

and

α > β

is the explicit formula for the

n

th Fibonacci number as given in this question.

The Golden Ratio is one of the roots of the quadratic equation and this explains the many connections between Fibonacci numbers and the Golden Ratio.

For a sequence of, mainly more elementary, problems on these topics see

See this article if you want to know more about the method of proof by mathematical induction.